Theoretically possible, which lead to a theoretical superluminal computer — a so-called hypercomputer.
From the link:
Physicists have come up with a way to process information faster than the speed of light. But what could they do with such a hypercomputer?
The speed of light represents one of the fundamental limits of the laws of physics. Nothing can travel faster than the speed of light, right?
Well, yes and no, say Volkmar Putz and Karl Svozil at the Vienna University of Technology in Austria. They say there are several ways that signals can cross the superluminal line, although none of them allow the kind of time travel paradoxes beloved of science fiction writers. For example, the quantum phenomenon of entanglement occurs when two quantum particles are described by the same wave function. These particles can be separated by the diameter of the universe and yet a measurement on one will instantaneously influence the other.
So-called “nonlocal” phenomenon cannot be used to transmit information faster than the speed of light but Putz and Svozil today ask whether it can be used to process it, to carry out computational tasks at superluminal speeds. They say there is no reason why not, provided the processing does not lead to any time travel paradoxes.
Assuming that this device could actually be built, what could you do with a superluminal computer? That’s a good question that Putz and Svozil do not address directly. They say such a device would fall into a class of processing machine known as hypercomputers. These are hypothetical devices more powerful than Turing machines, that allow non-Turing computations. They were first discussed by Alan Turing in the 1930s.
In theory, hypercomputers can compute certain kinds of otherwise noncomputable functions. That sounds handy but even though there are uncountably many non-computable functions, it’s actually quite hard to come up with an example of one that might seem useful. If you have any ideas, post them in the comments section.